87.22.13 problem 13

Internal problem ID [23758]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 4. The Laplace transform. Exercise at page 199
Problem number : 13
Date solved : Thursday, October 02, 2025 at 09:44:56 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+9 y&={\mathrm e}^{t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.055 (sec). Leaf size: 21

ode:=diff(diff(y(t),t),t)+9*y(t) = exp(t); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -\frac {\cos \left (3 t \right )}{10}-\frac {\sin \left (3 t \right )}{30}+\frac {{\mathrm e}^{t}}{10} \]

✓ Mathematica. Time used: 0.069 (sec). Leaf size: 27

ode=D[y[t],{t,2}]+9*y[t]==Exp[t]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{30} \left (3 e^t-\sin (3 t)-3 \cos (3 t)\right ) \end{align*}

✓ Sympy. Time used: 0.076 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-9*y(t) - exp(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {e^{3 t}}{12} - \frac {e^{t}}{8} + \frac {e^{- 3 t}}{24} \]